# Proof of Definite Integral

Let’s assume that $f$ is continuous and positive on the interval $[a, b]$. Then the definite integral $\int^b_a f(x) dx$ represents the area of the region bounded by the graph of $f$ and the x-axis, from x = a to x = b. First, we partition the interval $[a, b]$ into n subintervals, each of width $\Delta x = (b - a)/n$ such that $a = x_0 < x_1 < x_2 < . . . < x_n = b$ Then we can form a trapezoid for each subinterval and the area of the ith trapezoid = $[\frac{f(x_{i-1}) + f(x_i)}{2}](\frac{b-a}{n})$. This implies that the sum of the areas of the n trapezoids is Area = $\frac{b - a}{2n}[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)] = \frac{b - a}{2n}[f(x_0) + f(x_n) + 2\displaystyle\sum\limits_{i=1}^{n-1} f(x_i)] = \frac{b - a}{2n}(f(x_0) + f(x_n)) + \displaystyle\sum\limits_{i=1}^{n-1} f(x_i)(\frac{b - a}{n}) = \frac{b - a}{2n}(f(x_0) + f(x_n) - 2f(x_n)) + \displaystyle\sum\limits_{i=1}^{n} f(x_i)(\frac{b - a}{n}) - 2f(x_n)(\frac{b - a}{2n}) = \frac{b - a}{2n}(f(x_0) - f(x_n)) + \displaystyle\sum\limits_{i=1}^{n} f(x_i)\Delta x = \lim_{n\to\infty}\frac{b - a}{2n}(f(x_0) - f(x_n)) + \lim_{n\to\infty}\displaystyle\sum\limits_{i=1}^{n} f(x_i)\Delta x = 0 + \lim_{n\to\infty}\displaystyle\sum\limits_{i=1}^{n}f(x_i)\Delta x = \int^b_a f(x) dx$

# Exercise on Relations

Let S and S‘ be the following subsets of the plane: $S = \{(x,y) | y = x+1, 0 and $S'= \{(x,y) | y-x \in \mathbb{Z} \}$

a) Show that S’ is an equivalence relation on the real line and that $S \subset S'$.

Proof: Reflexivity $x-x \in \mathbb{Z}, \forall x \in \mathbb{R}$

Symmetry $z \in \mathbb{Z} \Rightarrow -z \in \mathbb{Z}$

Transitivity- If $x~y, y~z$ then $z-y=(z-x)-(x-y)$ and thus z-y is the difference of two integers which implies that z-y is itself an integer.

To show that $S \subset S'$ we note that $y=x+1 \Rightarrow y-x=1$ which $\in \mathbb{Z}$

b) Show that given any collection of equivalence relations on a set A, their intersection is an equivalence relation in A.

Proof: Let $\{R_\alpha\}_\alpha\in A$ be a nonempty class of equivalence relations and let $\Omega = \cap_{\alpha \in A}R_\alpha$

Reflexivity- If $(x,y) \in R_\alpha, \forall_\alpha \in A$ then $(x,y) \in \Omega \Rightarrow (y,x) \in R_\alpha, \forall_\alpha \in a \Rightarrow (y,x) \in \Omega$.

Symmetry $(x,x) \in R_\alpha, \forall_\alpha \in A \Rightarrow (x,x) \in \Omega$.

Transitivity– If $(x,y), (y,z) \in R_\alpha, \forall_\alpha \in A$ then $(x,y), (y,z) \in \Omega \Rightarrow (x,z) \in R_\alpha, \forall_{\alpha \in A} \Rightarrow (x,z) \in \Omega \therefore$ the intersection is an equivalence relation on A.